Canonical Models and Completeness

Canonical models for modal logics are constructed with possible worlds that are maximally consistent sets of formulas and with accessibility defined in a standard way. Logics for which canonical models suffice to establish ax­iomatic completeness are called canonical logics.

4.4.1 Canonical Modal Logics

Let KL be a normal modal logic, which we assume is axiomatized by adding a set of axioms to K. Call a set S of modal formulas KL-consistent provided

4.4 CanonicalModelsandCompleteness 61

The key fact about modal canonical models is the following. It is very well- known, but is included for reference.

Theorem 4.7 (Modal Truth Lemma)

A canonical model is a universal countermodel. For, suppose X is not prov­able in KL. Thenis KL-consistent and so can be extended to a maximally

consistent set Γ. Then Γ º G and, by the Truth Lemma,

What has been shown is that if X is not provable in KL, then X is falsifiable in the canonical model.

An axiomatic modal logic KL is canonical if its canonical model meets the semantic conditions adopted for KL. This is stated loosely because it requires an independent determination of what we want to count as a model for KL. The statement can be cleaned up, but we don’t need the resulting complica­tions here. We will simply assume the modal logics we discuss come with intended semantics. Then GL is not canonical, but most common modal logics are. Indeed, all the modal logics considered in Sections 2.7 and 4.3 are canon-

The key fact to take away is that canonical modal logics have axiomatic completeness proofs that are uniform across a broad range of modal logics.

4.4.2 Canonical Justification Models

Canonical models for justification logics were introduced in Fitting (2005). Until recently all known justification logics were counterparts of canonical modal logics, but recently Shamkanov (2016) has shown a realization theorem for the Godel-Ldb modal logic, and this is not canonical. Nonetheless, canoni­cal justification models supply the main semantic machinery behind almost all realization proofs, as we will see in Chapters 7 and 8.

Definition 4.8 (Consistency) Let JL(CS) be some axiomatically formulated justification logic, with constant specification CS.

JL(CS)-inconsistent.

We need an analog of the modal sharp operation, Definition 4.6, which was used in the previous section.

Definition 4.9 (Sharp Operation, Justification Version) For a set S of formu-

The following definition is directly analogous to that for modal canonical models—only part (4) is new.

Definition 4.10 (Canonical Justification Model) The canonical Fitting model

As with modal canonical models, the key item to show is a truth lemma, analogous to Theorem 4.7. Curiously, for justification logics the proof is sim­pler than in the modal case.

Theorem 4.11 (Justification Truth Lemma)

Proof The proof is by induction on the degree of X. The atomic case is by definition. Propositional connective cases are the same as in the modal proof for Theorem 4.7, making use of maximal consistency of Γ. This leaves the justification case. Assume as induction hypothesis that the result holds for for the formula X.

4.4.3 Strong Evidence and Fully Explanatory

In Definition 4.4 Fully Explanatory Fitting models were characterized, and in Definition 4.5 Strong Evidence Functions as well. As we are about to see, the second of these comes for free, while the first requires a familiar argument already seen in the proof of Theorem 4.7. We begin with Strong Evidence. Recall that an evidence function in a Fitting model is strong if the Evidence condition for it implies the Modal condition.

Theorem 4.12 (Strong Evidence) Let JL(CS) be an axiomatically presented

We move on to Fully Explanatory. Speaking informally, suppose we say a formula is knowable at a possible world of a Fitting model if it is true at all accessible possible worlds. Then Fully Explanatory says: Any formula that is knowable at a possible world has a justification at that possible world.

Theorem 4.13 (Fully Explanatory) Again let JL(CS) be an axiomatically presented justification logic, and letbe the canoni­

cal model for it. Assume JL has the internalization property relative to con­stant specification CS (by Theorem 2.14, this is the case if CS is axiomati- cally appropriate). Then the canonical model is fully explanatory. That is, if

4.5